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Metamath Proof Explorer

Theorem cxpne0d

Description: Complex exponentiation is nonzero if its base is nonzero. (Contributed by Mario Carneiro, 30-May-2016)

Ref Expression
Hypotheses cxp0d.1 ⊢ φ → A ∈ ℂ
cxpefd.2 ⊢ φ → A ≠ 0
cxpefd.3 ⊢ φ → B ∈ ℂ
Assertion cxpne0d ⊢ φ → A B ≠ 0

Proof

Step Hyp Ref Expression
1 cxp0d.1 ⊢ φ → A ∈ ℂ
2 cxpefd.2 ⊢ φ → A ≠ 0
3 cxpefd.3 ⊢ φ → B ∈ ℂ
4 cxpne0 ⊢ A ∈ ℂ ∧ A ≠ 0 ∧ B ∈ ℂ → A B ≠ 0
5 1 2 3 4 syl3anc ⊢ φ → A B ≠ 0